Among various recent proposals, a natural candidate of the longitudinal confining interaction is the collinear QCD [1]} in particular the 't Hooft model [2]}. Indeed, the 't Hooft model exhibits chiral symmetry breaking. Alas, the model is not analytically solvable. We instead propose an analytically solvable model for the longitudinal confining interaction that is closely related to the 't Hooft model, \(V_\Vert  = -\sigma ^2 \partial _x\big ( x(1-x) \partial _x \big )\) , where \(\sigma \)  is the strength of the confinement [3]}. \(\partial _x = (\partial /\partial x)|_{\vec{\zeta }_\perp }\)  can be identified with a recently introduced frame-independent coordinate \(\tilde{z} = (1/2)P^+x^-\)  [4]}.
Comparing to other form of longitudinal confinement, notably Refs. [5]}, [6]}, our proposal possesses several salient features [3]}. The solutions from this potential resemble the desired asymptotic parton distributions \(\sim x^a(1-x)^b\)  at the endpoints, which produces the GMOR relation in the vicinity of the chiral limit. In the heavy sector, the same interaction is shown to restore the 3D rotation symmetry.
Furthermore, with this interaction, the longitudinal Schrödinger equation (REF ) is analytically solvable [3]}.
For brevity, we will refer to this Hamiltonian as \(\mathrm {BLFQ_0}\) . The resulting full mass eigenvalues are,
\(M^2 = 2\kappa ^2(2n+|m|+J) + \sigma (m_q+m_{\bar{q}})(2l+1) \\+ \sigma ^2 l(l+1) + (m_q+m_{\bar{q}})^2,\) 
